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<title>Atlas software user guide -- K\G/B</title>
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<h2>K-orbits in G&#8260;B</h2>
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<i>Last updated: October 15, 2005</i>
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<p>
The classification of K-orbits in G&#8260;B is the basis of the geometric
picture of representation theory, where representations correspond to
K-equivariant local systems on such orbits. Also, the corresponding orbits
on the dual side provide the partition of representations into 
&#8220;L-packets&#8221; in Langlands' classification of representations.
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<p>
It turns out that the main combinatorial structure that the program uses
internally for the parametrization of <a href="blocks.html">blocks</a> of 
representations is in natural (1,1) correspondence with the set of K-orbits,
and moreover we can compute the conjugation action of W (a.k.a. the 
cross-action), and the Cayley transforms in this description. It would be 
possible to output actual orbit representatives as products of certain 
canonically defined elements of finite order in the group (or even as matrices,
for classical groups), but this is not currently implemented.
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<p>
In our language, K-orbits are in (1,1)-correspondence with
<a href="strongreal.html">strong involutions</a> normalizing T, in the 
G-conjugacy class corresponding to a strong real form lying over the chosen 
real form of G, modulo T-conjugacy. It is easy to see that this is equivalent 
to the description in [1].
The enumeration of K-orbits and the various actions are output by the
&#8220;kgb&#8221; command. For example, here is the output of &#8220;kgb&#8221;
for Sp(4,<b>R</b>) (the split form of simply connected C2):
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<pre>
 0:     1   2     6   4    0
 1:     0   3     6   5    0
 2:     2   0     *   4    0
 3:     3   1     *   5    0
 4:     8   4     *   *    1  2
 5:     9   5     *   *    1  2
 6:     6   7     *   *    1  1
 7:     7   6    10   *    2  212
 8:     4   9     *  10    2  121
 9:     5   8     *  10    2  121
10:    10  10     *   *    3  1212
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<p>
The first column is just the number of the orbit in the enumeration we are
using. The next two are the cross-actions for the two simple reflections in
the complex Weyl group; they are always defined. Then come two columns
describing the Cayley transforms for the simple reflections: they are defined
only if the simple root is imaginary noncompact for the corresponding
involution. The next-to-last column is the value of the length function;
in terms of orbits this is the difference in dimension between the current
orbit and the closed orbits; combinatorially, it is the smallest number of
operations (cross-actions or Cayley transforms) that will get you from
a closed orbit to the current one. Finally, the last column lists the
Weyl group element defining the root datum involution associated to the
orbit; in our language, this is just the restriction of the involution to
the torus. The Weyl group element is written as a product of simple 
reflections: for instance &#8220;121&#8221; means 
s<sub>1</sub>s<sub>2</sub>s<sub>1</sub>. The output is sorted by length, and
then by Weyl group element; the orbits associated to the identity element
in the Weyl group are the closed orbits&#8212;they are all isomorphic to
the flag variety of K.
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<p>
In the example, we are in the equal rank case, so root datum involutions are
just involutions in the Weyl group. There are four W-conjugacy classes of 
those, <i>viz.</i> {e}, {1,212}, {2,121}, {1212}, corresponding to the four 
<a href="cartan.html">conjugacy classes</a> of Cartan subgroups in 
Sp(4,<b>R</b>). They correspond to the four W-orbits for the cross-action: 
{0,1,2,3}, {4,5,8,9}, {6,7}, {10}. In this case, there is only one split strong
real form, and it is a <a href="strongreal.html">&#8220;packet&#8221;</a> all 
by itself. Hence the number of orbits lying over any given root datum 
involution is the cardinality of the component group of the torus dual to the 
corresponding Cartan subgroup. So there are four for the fundamental Cartan
(the dual torus is split), two for the intermediate Cartan that is 
<b>U</b>(1).<b>R</b><sup>&#215;</sup> (corresponding to reflections through
long roots), one for the intermediate Cartan that is <b>C</b><sup>&#215;</sup>
(corresponding to reflections through short roots), and one for the split
Cartan (the dual torus is compact.)
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<td>[1]</td>
<td>
R.W Richardson and T.A. Springer, The Bruhat Order on Symmetric Varieties,
Geom. Dedicata 35(1990), pp. 389-436.
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</tr>
</table>

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